Measurements of production and inelastic cross sections for $\mbox{p}+\mbox{C}$, $\mbox{p}+\mbox{Be}$, and $\mbox{p}+\mbox{Al}$ at 60 GeV/$c$ and $\mbox{p}+\mbox{C}$ and $\mbox{p}+\mbox{Be}$ at 120 GeV/$c$

This paper presents measurements of production cross sections and inelastic cross sections for the following reactions: 60 GeV/$c$ protons with C, Be, Al targets and 120 GeV/$c$ protons with C and Be targets. The analysis was performed using the NA61/SHINE spectrometer at the CERN SPS. First measurements were obtained using protons at 120 GeV/$c$, while the results for protons at 60 GeV/$c$ were compared with previously published measurements. These interaction cross section measurements are critical inputs for neutrino flux prediction in current and future accelerator-based long-baseline neutrino experiments.

This paper presents measurements of production cross sections and inelastic cross sections for the following reactions: 60 GeV=c protons with C, Be, Al targets and 120 GeV=c protons with C and Be targets. The analysis is performed using the NA61/SHINE spectrometer at the CERN Super Proton Synchrotron. First measurements are obtained using protons at 120 GeV=c, while the results for protons at 60 GeV=c are compared with previously published measurements. These interaction cross section measurements are critical inputs for neutrino flux prediction in current and future accelerator-based long-baseline neutrino experiments.

I. INTRODUCTION
Long-baseline neutrino beams are typically initiated by high-energy protons that strike a long target, yielding hadrons that can decay to neutrinos or can reinteract in the target (carbon and beryllium being the most frequently used materials) or in the aluminum focusing horns, potentially producing additional neutrino-yielding hadrons. The NA61/SPS Heavy Ion and Neutrino Experiment (NA61/SHINE) [1], which is a fixed-target experiment at the CERN Super Proton Synchrotron (SPS), has already been very successful at measuring the yields of secondary hadrons generated by protons at 31 GeV=c on carbon targets [2][3][4] for the T2K longbaseline neutrino oscillation experiment [5]. NA61/ SHINE has recently completed data collection at higher energies to benefit other accelerator-based long-baseline neutrino experiments, particularly experiments that use the NuMI beam line or the future LBNF beam line at Fermilab. NuMI is initiated by 120 GeV=c protons on a carbon target, while LBNF will use 60-120 GeV=c protons on a carbon target.
NA61/SHINE has already measured integrated cross sections of pions and kaons to constrain predictions of the neutrino flux coming from reinteractions of pions and kaons [6]. This paper presents measurements of proton integrated cross sections to further improve neutrino flux predictions coming from the primary interactions in the neutrino beam targets or reinteractions of protons in the target and aluminum horns.
During the 2016 data collection, NA61/SHINE recorded interactions of protons on thin carbon, beryllium, and aluminum targets using beam momenta of 60 and 120 GeV=c. Interactions were recorded with all three targets at 60 GeV=c, while interactions on thin carbon and beryllium targets were recorded at 120 GeV=c.
The methodology to measure the inelastic cross section σ inel and the production cross section σ prod follows the same approach as the previous NA61/SHINE measurements [6]. The inelastic process is defined as the sum of all stronginteraction processes that result in the disintegration of the target nucleus (including quasielastic interactions). This is equivalent to the total cross section minus the coherent elastic cross section. The production process is defined as that in which new hadrons are produced. Using the coherent elastic cross section, σ el , and the quasielastic cross section, σ qe , one can define σ inel and σ prod as It is worth noting that not all measurements and experiments use the same terminology for these processes. For instance, the MINERνA experiment [7] at NuMI uses the term "absorption" cross section for σ inel , while previous measurements sometimes refer to either σ prod or σ inel with the term absorption cross section (e.g., Carroll et al. [8] used σ prod as the absorption cross section, while Denisov et al. [9] used σ inel as the absorption cross section).

II. EXPERIMENTAL SETUP
NA61/SHINE receives a secondary hadron beam from the 400 GeV=c SPS proton beam. Upstream of the NA61/ SHINE detector, a magnet system is used to select the desired beam momentum between 13 and 350 GeV=c.
The NA61/SHINE detector [1] is shown in Fig. 1. It comprises two superconducting magnets, five time projection chambers (TPCs), a time-of-flight (TOF) system, and a forward hadron calorimeter (the Projectile Spectator Detector). Two of the TPCs, vertex TPC 1 (VTPC-1) and vertex TPC 2 (VTPC-2), are contained within superconducting magnets capable of generating a combined maximum bending power of 9 T · m. The most critical systems for integrated cross section measurements are the trigger system and the beam position detectors (BPDs). The trigger system uses two scintillator counters (S1 and S2) to trigger on beam particles and two annular scintillation counters (V0 and V1) to veto divergent beam particles upstream of the target. The 1 cm radius S4 scintillator sits downstream of the target and is used to determine whether or not an interaction has occurred.
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Funded by SCOAP 3 .
A Cherenkov differential counter with achromatic ring focus (CEDAR) [10,11] selects beam particles of the desired species. For the 2016 data at 60 GeV=c (120 GeV=c), the beam was composed of approximately 22% (40%) protons.
Beam particles are selected by defining the beam trigger (T beam ) as the coincidence of S1 ∧ S2 ∧ V0 ∧ V1 ∧ CEDAR. The interaction trigger (T int ) is defined by the coincidence of T beam ∧ S4 to select beam particles which have interacted with the target. A correction factor for interactions that result in an S4 hit will be discussed in detail in Sec. VA. Three BPDs, which are proportional wire chambers, are located 30.39, 9.09, and 0.89 m upstream of the target and determine the trajectory of the incident beam particle to an accuracy of approximately 100 μm.
Two types of carbon targets were used: one composed of graphite of a density of ρ ¼ 1.84 g=cm 3 with dimensions of 25 mm ðWÞ × 25 mm ðHÞ × 20 mm ðLÞ for the 60 GeV=c proton beam, corresponding to roughly 4.2% of a protonnuclear interaction length, and one composed of graphite of a density of ρ ¼ 1.80 g=cm 3 with dimensions of 25 mm ðWÞ × 25 mm ðHÞ × 14.8 mm ðLÞ for the 120 GeV=c proton beam, corresponding to roughly 3.1% of a proton-nuclear interaction length. The former is the same graphite target as was used for past NA61/SHINE measurements, while the latter is a newly produced target using the same type of graphite as the NuMI target. The beryllium target has a density of ρ ¼ 1.85 g=cm 3 with dimensions of 25 mm ðWÞ × 25 mm ðHÞ × 14.9 mm ðLÞ, corresponding to roughly 3.5% of a proton-nuclear interaction length. This beryllium target is a newly produced target. The aluminum target has a density of ρ ¼ 2.70 g=cm 3 with dimensions of 25 mm ðWÞ × 25 mm ðHÞ × 14.8 mm ðLÞ, corresponding to roughly 3.6% of a proton-nuclear interaction length. This aluminum target is the same target as was used for past NA61/SHINE measurements.

III. EVENT SELECTION
Several cuts were applied to events to ensure the purity of the samples and to control the systematic effects caused by beam divergence. First, the so-called WFA (wave form analyzer) cut was used. The WFA determines the timing of beam particles that pass through the S1 scintillator. If another beam particle passes through the beam line close in time, it could cause a false trigger in the S4. In order to mitigate this effect, a conservative cut of AE2 μs was applied, ensuring that only one particle is allowed to pass through the S1 in a 4 μs time window.
Beam trajectory measurements are especially important for estimating the effects of beam divergence. To understand these effects, tracks are fitted to the reconstructed BPD clusters, and these tracks are extrapolated to the S4 location. The so-called "good BPD" cut requires that the event includes a cluster in the most-downstream BPD and that a track was successfully fit to the BPDs. Figure 2 shows examples of the resulting BPD extrapolation to the S4. As seen in Fig. 2 (left), a halo of beam particles can miss the S4, mimicking the interaction trigger. To avoid such an effect and also to minimize the effect of the S4 size and position uncertainties, which will be discussed in Sec. VI, a radial cut of 0.75 cm was applied to the tracks extrapolated from the BPDs, as indicated in Fig. 2. After the p þ C60 GeV=c data collection, the S4 position was realigned for other measurements which can also be seen in Fig. 2.
About two-thirds of the data were collected with the target inserted and one-third of the data were collected with the target removed. The number of events remaining after the described selection cuts for the target inserted and removed are shown in Tables I-3 for C, Be, and Al.

IV. INTERACTION TRIGGER CROSS SECTIONS
The probability of a beam particle interaction inside a thin target is proportional to the thickness, L, and the number density of the target nuclei, n, in the thin target approximation. Thus, the interaction probability, P, can be defined in terms of the interaction cross section, σ: The counts of beam and interaction triggers as described in Sec. II can be used to estimate the trigger probability as follows: where NðT beam Þ is the number of beam events passing the event selection cuts and NðT beam ∧ T int Þ is the number of selected beam events that also have an interaction trigger. In order to correct for events in which the beam particle interacts outside of the target, such as interactions on beam line materials or air, data were also recorded with the target removed from the beam. Table IV summarizes the trigger probabilities for both the target inserted (I) and removed (R) data.   Taking into account the trigger probabilities with the target inserted and removed, P I Tint and P R Tint , the interaction probability P int can be obtained as Using Eqs. (3)-(5), the trigger cross section, σ trig , can be written as where N A , ρ, and m A are Avogadro's number, the material density, and the atomic mass. The detailed calculation is described in Ref. [6].

A. S4 trigger correction factors
The trigger cross section comprises interactions where the resulting particles miss the S4 scintillator. But even when there has been an interaction in the target, there is a possibility that a forward-going particle will strike the S4. Moreover, not all elastically scattered beam particles strike the S4. Corrections must be applied to account for these effects. From Eqs. (1) and (2), the trigger cross section can be related to the production and inelastic cross sections with correction factors: and Here, f prod , f qe , and f el are the fractions of production, quasielastic, and elastic events that miss the S4 counter. These correction factors, as well as σ qe and σ el , are estimated from Monte Carlo (MC) simulations. GEANT4 version 10.4.p03 [12][13][14] was used to estimate the MC correction factors. The QBBC physics list was used to estimate correction factors relating to elastic events, while the FTFP_BERT physics list was used to estimate correction factors for other events. The resulting MC correction factors are summarized in Table V (nominal).

B. Beam purity
Kaons are the most probable source of contamination for proton beams. In the case of proton beams at 60 and 120 GeV=c, the CEDAR detector has enough power to discriminate protons from other charged particles. The upper limit on kaon contamination was found to be smaller than 0.1% at 120 GeV=c from pressure scans taken of the CEDAR detector and even lower at 60 GeV=c. It was concluded that the beam purity has a negligible impact on integrated cross section measurements and no correction factor was applied.

A. Target density
The uncertainty on the target density affects the calculation of σ trig as shown in Eq. (6). The density uncertainty for each target was estimated by calculating the standard deviation of the target densities determined from measurements of the mass and dimensions of the machined target samples. (There were several machined samples fabricated for each target type.) This evaluation led to a 0.69% uncertainty on carbon, 0.19% uncertainty on beryllium, and a 0.29% uncertainty on aluminum, respectively.

B. S4 size and position
Another systematic uncertainty comes from the size and position of the S4 scintillator. The diameter of the S4 has previously been found to have an uncertainty of AE0.40 mm. The S4 position has been determined using BPD tracks extrapolated to the S4 location. A conservative S4 position uncertainty of AE1.0 mm in X and Y coordinates is assigned. In order to propagate these uncertainties to σ inel and σ prod , two additional MC samples with the S4 TABLE V. Correction factors to the nominal MC simulation for the elastic process obtained with QBBC, and for other processes obtained with FTFP_BERT. The right-hand side shows ratios to the nominal MC simulation for a sample used to assess systematic uncertainties; these ratios were obtained using FTFP_BERT for the elastic process and FTF_BIC for other processes. Model uncertainty treatment is further discussed in Sec. VI C.

MC correction factors (nominal)
Ratio to nominal (systematic) diameter modified and four additional MC samples with the S4 position shifted were generated. Previous NA61/SHINE analyses have found that S4 inefficiency is negligibly small [2,6] and this analysis also used the same S4 scintillator. The S4 inefficiency is concluded to be less than 0.1% and neither an uncertainty nor a correction relating to the S4 scintillator efficiency is applied to the results.

C. Model uncertainties
Physics model uncertainties on the S4 trigger correction factors were estimated for elastic and other processes separately. GEANT4 version 10.4.p03 has two models for the elastic process: Barashenkov-Glauber-Gribov and Chips. The former is available with the QBBC physics list, is used for the nominal correction, and is the recommended model by GEANT4. The latter is available with other physics lists including FTFP_BERT. In order to estimate the model uncertainties associated with the elastic process, the S4 correction factors f el and σ el were recalculated with FTFP_BERT, and ratios to the nominal MC simulation are shown in Table V (systematic). Additionally, validity of the model uncertainties on σ el for p þ C at 60 and 120 GeV=c have been evaluated with former σ el measurements by Bellettini et al. at 21.5 GeV=c [15] and Schiz et al. at 70 GeV=c [16] and found to be consistent within uncertainty.
The S4 correction factors f prod , f inel , and f qe as well as σ qe were estimated with FTFP_BERT. In order to estimate the model uncertainties associated with these correction factors, the correction factors were recalculated with three additional physics lists: QBBC, QGSP_BERT, and FTF_BIC. Using these additional correction factors, the model dependence of the integrated cross section measurements was studied. As an example, ratios to the nominal MC simulation obtained with FTF_BIC are shown in Table V (systematic). All systematic uncertainties discussed in this section are summarized in Tables VI and VII for production and inelastic cross section measurements.

VII. RESULTS AND DISCUSSION
Several production cross sections have been measured in this analysis. Statistical, systematic, and physics model   Table VIII. Production cross sections were measured to be higher than the predictions of GEANT4. The p þ C and p þ Al at 60 GeV=c measurements are compared with the results by Carroll et al. [8] as shown in Fig. 3 (left). The new NA61/SHINE results are consistent within errors with the previous measurements, and our statistical and systematic uncertainties are smaller. Several inelastic cross sections have also been determined in this analysis. Statistical, systematic, and physics model uncertainties were estimated separately and are summarized in Table IX. For comparison, ratios to the GEANT4 10.4.p03 predictions with FTFP_BERT are also shown in Table IX. Inelastic cross sections were measured to be higher than the predictions of GEANT4. The measurements with 60 GeV=c protons are compared with the results by Denisov et al. [9] in Fig. 3 (right). The measurements of p þ C and p þ Al at 60 GeV=c are found to be consistent within errors, while the p þ Be at 60 GeV=c inelastic cross section is found to be slightly lower by about 1 standard deviation.
For the proton beam at 120 GeV=c, large GEANT4 physics model dependences were observed. This is due to differences between the correction factors predicted by different physics list, and in particular from FTF_BIC, which has large differences from other physics lists. Differences in these values compared to the nominal values in Table V cause large model uncertainties on nonelastic processes. One possible reason is that the size and position of the S4 scintillator was not optimal for a 120 GeV=c beam. Furthermore, future direct measurements of quasielastic processes will help to reduce model uncertainties, since the measurements presented in this paper have achieved a few % level statistical and systematics uncertainties.

VIII. SUMMARY
In summary, production and inelastic cross sections of protons on carbon, beryllium, and aluminum targets were measured.
The production cross section with a proton beam at 120 GeV=c was measured for the first time with a precision of about 6% (8%) for p þ C (p þ Be) including statistical, systematic, and model uncertainties. At 60 GeV=c, the measured production cross sections were comparable to previous results for p þ C and p þ Al, and the precision was improved to about 3%. The production cross section of p þ Be at 60 GeV=c was measured for the first time with a precision of about 4% including statistical, systematic, and model uncertainties.
The inelastic cross section with a proton beam at 120 GeV=c was measured for the first time with a precision of about 6% (8%) for p þ C (p þ Be) including statistical, systematic, and model uncertainties. For the inelastic  production cross section of the proton beam at 60 GeV=c, reasonable agreement with a previous measurement was found. The current uncertainties on NuMI and LBNF beam predictions have to extrapolate from data at lower or higher energy than the actual beam energy. Thus, new measurements presented in this paper will improve flux predictions by removing the necessity to extrapolate from different energies.